Circle Circumference
Description
The circumference is the linear distance around the outside edge of a circle. It is essentially the "perimeter" of the circle. The formula $C = 2\pi r$ (or $C = \pi d$) connects this curved length directly to the circle's width (diameter) or radius.
This relationship implies that for any circle, no matter how large or small, the ratio of its circumference to its diameter is always the same constant number: $\pi$ (approximately 3.14159).
Visually, if you were to cut a circle open and unroll it flat, the length of that straight line would be the circumference.
History & Origins
The discovery of the ratio between a circle's circumference and its diameter is one of the earliest mathematical achievements. Babylonians (c. 1900 BC): Approximated $\pi$ as 3.125. Ancient Egyptians (c. 1650 BC): The Rhind Papyrus gives a value equivalent to 3.16. Archimedes (c. 250 BC): He was the first to rigorously approximate $\pi$ by inscribing and circumscribing polygons with many sides around a circle. He determined that the value lies between $3\frac{10}{71}$ and $3\frac{1}{7}$. Zu Chongzhi (c. 480 AD): A Chinese mathematician who calculated $\pi$ to seven decimal places (3.1415926) using a 12,288-sided polygon, a record that stood for 800 years.
Limit of a Regular Polygon
We can approximate a circle as a regular polygon with an infinite number of sides.
Consider a regular $n$-sided polygon inscribed in a circle of radius $r$.
Divide the polygon into $n$ isosceles triangles with central angle $\theta = \frac{360^\circ}{n}$.
The base of each triangle (a side of the polygon) has length $s = 2r \sin(\frac{180^\circ}{n})$.
The total perimeter of the polygon is $P_n = n \times s = 2nr \sin(\frac{180^\circ}{n})$.
As $n$ approaches infinity (more and more sides), the polygon becomes a circle.
Using the limit $\lim_{x \to 0} \frac{\sin x}{x} = 1$ (where $x$ is in radians), we can show that as $n \to \infty$, $P_n \to 2\pi r$.
Variables
| Symbol | Meaning |
|---|---|
C | Circumference (Perimeter) |
r | Radius (Distance from center to edge) |
π | Pi (approx. 3.14159) |
d | Diameter (2r) |
Examples
Basic Calculation
Problem: Find circumference with r=7
Solution:
Bicycle Odometer
Problem: A bicycle wheel has a diameter of 70 cm. How far does the bike travel in one complete revolution of the wheel?
Solution: ~2.2 meters
- Identify Diameter: $d = 70$ cm.
- Formula: $C = \pi d$ (since $2r = d$).
- Substitute: $C = \pi \times 70$.
- Calculate: $C \approx 3.14159 \times 70 \approx 219.91$ cm.
- Convert to meters: $2.199$ meters.
Running Track
Problem: A circular running track has a radius of 50 meters. If a runner does 10 laps, how far did they run?
Solution: ~3.14 km
- Calculate circumference of one lap: $C = 2\pi(50) = 100\pi$.
- Value for one lap: $100 \times 3.14159 \approx 314.16$ meters.
- Multiply by 10 laps: $314.16 \times 10 = 3141.6$ meters.
- Convert to km: $3.14$ km.
Common Mistakes
Confusing Area and Circumference
Remember: Area is the space INSIDE ($A = \pi r^2$, units squared). Circumference is the distance AROUND ($C = 2\pi r$, linear units).
Mixing up Radius and Diameter
Check if the problem gives you radius ($r$) or diameter ($d$). Remember $d = 2r$. If you use diameter in the $2\pi r$ formula without dividing by 2, your answer will be double what it should be.
Real-World Applications
Mechanical Engineering (Gears)
The circumference determines the gear ratio and travel distance. If a gear with a circumference of 10cm turns a gear with a circumference of 20cm, the second gear turns exactly half as fast.
Earth's Circumference
Eratosthenes (c. 240 BC) famously calculated the Earth's circumference by measuring shadow angles in two different cities. He used simple geometry to estimate it was about 40,000 km, which is incredibly close to the modern value (40,075 km).
Frequently Asked Questions
What is the difference between pi and 3.14?
Pi ($\pi$) is an irrational number with infinite decimal places. 3.14 is just a rounded approximation. Using $\pi$ is always more accurate.
Is perimeter the same as circumference?
Yes. "Perimeter" is the general term for the distance around any shape. "Circumference" is the specific name for the perimeter of a circle.